Large data limit of the MBO scheme for data clustering: $\Gamma$-convergence of the thresholding energies

TL;DR

This paper rigorously analyzes the large data limit of the MBO scheme for data clustering, showing convergence of the scheme's minimizers to solutions of a weighted optimal partition problem.

Contribution

It establishes the convergence of the MBO scheme's minimizers to a weighted optimal partition problem in the large data limit, providing a rigorous theoretical foundation.

Findings

MBO scheme minimizers converge to weighted optimal partitions

The scheme is consistent with the gradient descent of thresholding energies

Analysis applies to similarity graphs in data clustering

Abstract

In this work we begin to rigorously analyze the MBO scheme for data clustering in the large data limit. Each iteration of the MBO scheme corresponds to one step of implicit gradient descent for the thresholding energy on the similarity graph of some dataset. For a subset of the nodes of the graph, the thresholding energy at time $h$ is the amount of heat transferred from the subset to its complement at time $h$, rescaled by a factor $\sqrt{h}$. It is then natural to think that outcomes of the MBO scheme are (local) minimizers of this energy. We prove that the algorithm is consistent, in the sense that these (local) minimizers converge to minimizers of a suitably weighted optimal partition problem.